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Solve x² + 4x + 3 = 0

The solutions of x² + 4x + 3 = 0 are x = -1 or x = -3.

x² +x += 0
x2+4x+3=0⇒x=−1, −3x^2 + 4x + 3 = 0 \Rightarrow x = -1,\ -3
Discriminant D = 4 · x ≈ -1, -3

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = x² + 4x + 3-5-4-3-2-11-224681012x = -1x = -3vertexy = x² + 4x + 3
Graph of y = x² + 4x + 3. The curve crosses the x-axis at x = -1 and x = -3; the vertex is at (-2, -1).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = 4, c = 3.

    x2+4x+3=0x^2 + 4x + 3 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=42−4⋅1⋅3=16−12=4D = 4^2 - 4\cdot1\cdot3 = 16 - 12 = 4
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (x+1)(x+3)=0\left(x + 1\right)\left(x + 3\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x+1=0⇒x=−1x+3=0⇒x=−3x + 1 = 0 \Rightarrow x = -1\qquad x + 3 = 0 \Rightarrow x = -3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−4±42x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{4}}{2}
  7. 7

    Simplify to get the solutions:

    x=−1orx=−3x = -1\quad\text{or}\quad x = -3
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = 1·(1) + 4·(-1) + 3 = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 + 4x + 3 = 0\ \Rightarrow\ x = -1,\ -3 \]
\[ x^2 + 4x + 3 = 0 \]
\[ D = 4^2 - 4\cdot1\cdot3 = 16 - 12 = 4 \]
\[ \left(x + 1\right)\left(x + 3\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{4}}{2} \]
\[ x = -1\quad\text{or}\quad x = -3 \]

The parabola y = x² + 4x + 3

xy
-58
-43
-30
-2-1
-10
03
18

Questions

What are the solutions of x² + 4x + 3 = 0?

The solutions of x² + 4x + 3 = 0 are x = -1 or x = -3.

Does x² + 4x + 3 = 0 factor?

Yes: (x+1)(x+3) = 0.

What is the vertex of y = x² + 4x + 3?

(-2, -1). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -4, product = c/a = 3.

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